The roots of the equation `x^(3) -2x^(2) -x +2 =0` are

Options

A. 1,2,3
B. `-1,1,2`
C. `-1,0,1`
D. ` -1,-2,3`

Answer

To find the roots of the equation \( x^3 - 2x^2 - x + 2 = 0 \), we can follow these steps:

### Step 1: Identify the polynomial
Let \( f(x) = x^3 - 2x^2 - x + 2 \).

### Step 2: Use the Rational Root Theorem
We will test possible rational roots using the Rational Root Theorem. The possible rational roots can be the factors of the constant term (2) divided by the factors of the leading coefficient (1). The possible rational roots are \( \pm 1, \pm 2 \).

### Step 3: Test the possible roots
1. **Test \( x = 1 \)**:
\[ f(1) = 1^3 - 2(1^2) - 1 + 2 = 1 - 2 - 1 + 2 = 0 \]
Since \( f(1) = 0 \), \( x = 1 \) is a root.

2. **Test \( x = -1 \)**:
\[ f(-1) = (-1)^3 - 2(-1)^2 - (-1) + 2 = -1 - 2 + 1 + 2 = 0 \]
Since \( f(-1) = 0 \), \( x = -1 \) is also a root.

3. **Test \( x = 2 \)**:
\[ f(2) = 2^3 - 2(2^2) - 2 + 2 = 8 - 8 - 2 + 2 = 0 \]
Since \( f(2) = 0 \), \( x = 2 \) is also a root.

### Step 4: Factor the polynomial
Since we have found three roots \( x = 1, -1, 2 \), we can express the polynomial as:
\[ f(x) = (x - 1)(x + 1)(x - 2) \]

### Step 5: Conclude the roots
Thus, the roots of the equation \( x^3 - 2x^2 - x + 2 = 0 \) are:
\[ x = 1, -1, 2 \]

### Final Answer
The roots of the equation are \( x = 1, -1, 2 \).

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